Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?

The general element is $\pm\exp(A)$ where $A$ is skewsymmetric. (This gives each element infinitely often). This trick essentially works for all compact Lie groups. There is also the Cayley parameterization: $(I+A)(IA)^{1}$ for skewsymmetric $A$ is the general element of $SO(3)$ which lacks an eigenvalue $1$ (so isn't a halfturn.) This parameterizes all such matrices once each. 


First, if $A\in O(3)$ then either $A$ or $A$ is in $SO(3)$ so you can just restrict attention to $SO(3)$. For that, one answer is the Cayley transform: let $so(3)$ denote the set of $3\times 3$ antisymmetric matrices (which is homeomorphic to $\mathbb{R}^3$) and put $U=\{R\in SO(3) : \det(R+I)\neq 0\}$ Then $U$ is a dense open subset of $SO(3)$ and there is a homeomorphism $f:so(3)\to U$ given by $f(A)=(I+A)(IA)^{1}$. Another answer is to use the double cover $g:S^3\to SO(3)$ given by the conjugation action unit quaternions on purely imaginary quaternions. There is an explicit formula here: http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Conversion_to_and_from_the_matrix_representation 


I assume that you are interested in the case of matrices with real entries. Perhaps one should reduce the problem to $SO(3)$, since $O(3)$ has two connected components, or we may use the determinant $\pm1$ as one parameter. A natural parametrization is unlikely, or will not be injective, because $SO(3)$ has a nontrivial topology: its fundamental group is $\mathbb Z/2\mathbb Z$. One possibility is to give a direction $\vec u$ (a unitary vector) of the axis of rotation, and the angle $\theta\in\mathbb R/2\pi\mathbb Z$ of this rotation. But then$(\vec u,\theta+\pi)\sim(\vec u,\theta)$, and all pairs $(\vec u,0)$ yield the same element $I_3$. 


If there were a really simple way we wouldn't need the concept of "gimbal lock" (http://en.wikipedia.org/wiki/Gimbal_lock). In other words the manifold in question is compact but isn't the 3torus, and pretending it is has to break down somewhere. The unit quaternions are the "slickest" surrogate, but "parametrize" implies charts and global charts aren't going to work out "simply". 

